On the supercritical mean field equation on pierced domains

Abstract

We consider the problem $$(P_\eps)\qquad Δu +λ{e^{u}\over\int\limits_{Ω\setminus B(ξ,\eps)}e^u}=0\ \hbox{in}\ Ω\setminus B(ξ,\eps),\quad u =0\ \hbox{on}\ \partial\(Ω\setminus B(ξ,\eps)\), $$ where $\Omega$ is a smooth bounded open domain in $\rr^2$ which contains the point $\xi .$ We prove that if $\lambda >8\pi ,$ problem $(P_\eps)$ has a solutions $u_\eps$ such that $$u_\eps(x)\to {8π+ λ\over2} G(x,ξ) \ \hbox{uniformly on compact sets of $Ω\setminus\{ξ\}$}$$ as $\eps$ goes to zero. Here $G$ denotes Green's function of Dirichlet Laplacian in $\Omega .$ If $\lambda \notin 8\pi \mathbb{N}$ we will not make any symmetry assumptions on $\Omega ,$ while if $\lambda \in 8\pi \mathbb{N}$ we will assume that $\Omega$ is invariant under a rotation through an angle ${8π^2\over \la} $ around the point $\xi .$

References in corpus (1)

• Existence and uniqueness for Mean Field Equations on multiply connected domains at the critical parameter